Prof. P. E. Chase on Planetary Interaction. 507 



all star-masses at a stage of their development exhibit a phase 

 of volcanic activity. And it is possible that during this period 

 many of those masses which are of comparatively trifling 

 magnitude may have been partially or wholly broken up and 

 resolved into small fragments. 



LX. On the Nebular Hypothesis. — II, Interaction. By Pliny 



Eaele Chase, Professor of Philosophy in Haverford College. 



[Continued from p. 319.] 



THE tendency of vibrations in elastic media to produce har- 

 monic vibrations, combined with the tendency to geo- 

 metrical and harmonic as well as to arithmetical progressions, 

 should prepare us to look for relations of various complexity 

 when we study the mutual actions and reactions of planetary 

 masses. 



The triangular numbers, 1, 1 + 2, 1 + 2 + 3, 1 + 2 + 3 + 4, ap- 

 pear as indices in the following equation among powers of the 

 masses of the five principal bodies in the solar system : — 



(IHfMiHD 



This appears to be the first equation ever discovered which 

 introduces the products of powers of masses in so systematic 

 a form*. Although its full interpretation may be at present 

 beyond our reach, we can catch glimpses of its meaning, and 

 we may feel a reasonable assurance that it represents some im- 

 portant functional law of equilibrating tendencies between 

 centripetal and centrifugal forces. The truth of the equation 

 is, of course, independent of any assumption with regard to the 

 proper unit of comparison; but the dominance of the solar influ- 

 ence lends interest to the aesthetic harmony afforded by its intro- 

 duction. This interest is increased by the accordance between 

 the order of position and the magnitude of the indices in the 

 left-hand member of the equation, and by the fact that the 

 nebular centre of planetary inertia ( \Z%mr 2 -i-'Zinr) is in Sa- 

 turn's orbit. 



There is still some uncertainty as to the masses of Neptune 

 and Uranus; so that it is impossible to tell how close this agree- 

 ment may be; but the deviation from precise accuracy cannot 

 be large. According to Newcomb's latest determinations of 



* Laplace, however (Mec. Cel. II. vol. viii. pp. 65-69; VI. vol. ii. pp. 

 12-16, &c), investigated inequalities depending on squares and products 

 of the disturbing forces. In his discussions of the Jovian and Saturnian 

 systems he introduced terms containing the third and fifth dimensions of 

 eccentricities and inclinations. 



